A continuous linear operator T, on the space of entire functions in d variables, is PDE-preserving for a given set

of polynomials if it maps every kernel-set ker P(D),

, invariantly. It is clear that the set

of PDE-preserving operators for

forms an algebra under composition. We study and link properties and structures on the operator side

versus the corresponding family

of polynomials. For our purposes, we introduce notions such as the PDE-preserving hull and basic sets for a given set

which, roughly, is the largest, respectively a minimal, collection of polynomials that generate all the PDE-preserving operators for

. We also describe PDE-preserving operators via a kernel theorem. We apply Hilbert's Nullstellensatz.